- #1
hiigaranace
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Homework Statement
Find the point closest to the origin on the line of intersection of the planes y + 2z = 12 and x + y = 6
Homework Equations
[tex]\nu[/tex]f = [tex]\lambda\nu[/tex]g1 +[tex]\mu\nu[/tex]g2
f = x2+y2+z2
g1: y + 2z = 12
g2: x + y = 6
There are supposed to be gradients on all of those, whether or not LaTeX wants to show them.
The Attempt at a Solution
Let [tex]\nu[/tex]f(x, y, z) = 2x[tex]\vec{i}[/tex]+2y[tex]\vec{j}[/tex]+2z[tex]\vec{k}[/tex], [tex]\nu[/tex]g1(x, y, z) = [tex]\vec{j}[/tex]+2[tex]\vec{k}[/tex], and [tex]\nu[/tex]g2(x, y, z) = [tex]\vec{i}[/tex]+[tex]\vec{j}[/tex]
this gives:
2x = [tex]\mu[/tex] 2y = [tex]\lambda + \mu[/tex] 2z = [tex]2\lambda[/tex]
I tried pushing ahead from here, but I end up getting nowhere. Can someone please help me?
...annnnnnnnnnnd much as I hate to admit it, I'm having a lot of trouble with lagrange multipliers in the first place, and my textbook is sadly not a whole lot of help. If anyone out there can explain how to work these out in a general sense, I would very much appreciate it.
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